Chris Chiasson <chris.chiasson at gmail.com> wrote:
> I don't know about other version, but the following makes me worried:
> In[1]:=
> Limit[ArcTan[1/x],x->0]==Limit[ArcTan[1/x],x->0,Direction->-1]
> Out[1]=
> True
> In[2]:=
> Limit[ArcTan[1/x],x->0]==Limit[ArcTan[1/x],x->0,Direction->1]
> Out[2]=
> False
> In[3]:=
> $Version
> Out[3]=
> 5.0 for Microsoft Windows (November 18, 2003)
>
> Isn't there a requirement that the right and left limits match before
> a limit exists?
In mathematics, for the bilateral limit to exist, yes.
But Mathematica has no bilateral (or omnidirectional) limit. The
documentation is deceptive (although, of course, not purposefully so).
This regrettable matter has been brought up repeatedly in this newsgroup.
Here's a link to a response by Daniel Lichtblau in one such thread:
<http://groups-beta.google.com/group/comp.soft-sys.math.mathematica/msg/25c0f5767d75ef2f?hl=en>
Of course, it will continue to be brought up until adequately noticeable
documentation is provided.
In any event, note that
In[100]:=
Options[Limit]
Out[100]=
{Analytic -> False, Assumptions -> $Assumptions, Direction -> Automatic}
In[101]:=
?Direction
Direction is an option for Limit. Limit[expr, x -> x0, Direction -> 1]
computes the limit as x approaches x0 from smaller values. Limit[expr, x ->
x0, Direction -> -1] computes the limit as x approaches x0 from larger
values. Direction -> Automatic uses Direction -> -1 except for limits
at Infinity, where it is equivalent to Direction -> 1.
But even that documentation is still incomplete: What is the default
direction if x0 happens not to lie on the extended real line?
David Cantrell